Miller, Kierste – Homework 2 – Due: Sep 6 2007, 3:00 am – Inst: JEGilbert
2
2.
Moves once clockwise along the circle
(5
x
)
2
+ (3
y
)
2
= 1
,
starting and ending at (0
,
3).
3.
Moves once counterclockwise along the
ellipse
x
2
25
+
y
2
9
= 1
,
starting and ending at (0
,
3).
4.
Moves along the line
x
5
+
y
3
= 1
,
starting at (5
,
0) and ending at (0
,
3).
5.
Moves along the line
x
5
+
y
3
= 1
,
starting at (0
,
3) and ending at (5
,
0).
6.
Moves once counterclockwise along the
circle
(5
x
)
2
+ (3
y
)
2
= 1
,
starting and ending at (0
,
3).
Explanation:
Since
cos
2
t
+ sin
2
t
= 1
for all
t
, the particle travels along the curve
given in Cartesian form by
x
2
25
+
y
2
9
= 1 ;
this is an ellipse centered at the origin.
At
t
= 0, the particle is at (5 sin 0
,
3 cos 0),
i.e.
,
at the point (0
,
3) on the ellipse.
Now as
t
increases from
t
= 0 to
t
=
π/
2,
x
(
t
) increases
from
x
= 0 to
x
= 5, while
y
(
t
) decreases from
y
= 3 to
y
= 0 ; in particular, the particle
moves from a point on the positive
y
-axis to
a point on the positive
x
-axis, so it is moving
clockwise
.
In the same way, we see that as
t
increases
from
π/
2 to
π
, the particle moves to a point
on the negative
y
-axis, then to a point on the
negative
x
-axis as
t
increases from
π
to 3
π/
2,
until Fnally it returns to its starting point on
the positive
y
-axis as
t
increases from 3
π/
2 to
2
π
.